>>>1. How important is having other web sites link to you, and vice versa, in getting ranked high in the engines?

Many search engines are using link popularity to rank pages. It is very important to have incoming links. These links should be from sites that are relevant. If your site is about cars, then don't get a pet supply site to link to you. It will not help. FFA's are useless for this purpose.

>>>2. How important are header tags (ie. <h1></h1>) in getting ranked higher?

Some search engines are still showing some preference to <h1> tags. I don't know how long that will last, but it can't hurt to use them.

>>>3. How important is having links on your page that lead to pages named keyword.html for getting ranked higher (keyword.html being whatever keywords you are targeting)?

I am not sure it is important to have links leading to "keyword.html". It is important to have your site easy to navigate, and it can't hurt to use keywords in your page names.